Method and system for signal procesing non-uniformly sampled data

ABSTRACT

According to various embodiments, a method is provided for improving data and a system is provided that is configured to perform the method. The method can comprise processing a data signal by using an optical system comprising a signal processor. The method can comprise collecting data generated by the optical system, wherein the data comprises non-uniformly sampled data. The method can comprise performing an interpolation operation on the non-uniformly sampled data using the signal processor, to generate interpolated data. Further, the method can comprise adjusting the data with the interpolated data using the signal processor, to generate improved data. The improved data can be output to a user, for example, by displaying the improved data on a display unit, or by printing out the improved data. According to various embodiments, the data can comprise any desired data, for example, image data. The method can comprise improving the image resolution, improving the image brightness, improving the image contrast, and/or improving the image focus.

STATEMENT OF GOVERNMENT INTEREST

The invention described herein was made by an employee of the UnitedStates Government and may be manufactured and used by or for theGovernment of the United States of America for governmental purposeswithout the payment of any royalties thereon or therefor.

FIELD

The present invention relates generally to digital signal processing,and more particularly to a more effective manner to interpolate datafrom non-uniformly sampled data.

BACKGROUND

In the field of signal processing methods are commonly used to transformdata between different domains, for example, from the time domain to thefrequency domain. One such equation that can be used to transform datais the Whittaker-Shannon-Kotelnikov (“WSK”) sampling theorem. Thetheorem describes two processes in signal processing, a sampling processin which a continuous time signal is converted into a discrete timesignal, and a reconstruction process in which the original continuoustime signal is recovered from the discrete signal. Reconstruction of theoriginal continuous time signal is an interpolation process thatmathematically defines a continuous-time signal from the discretesamples and times between the sample instants. During the interpolationprocess, unknown data values are approximated from surrounding knowndata values.

The WSK sampling theorem corresponds to Nyquist sampled data. The WSKsampling theorem provides a method for reconstructing a continuous timeband-limited function from a discrete set of data points. The WSKsampling theorem assumes constant intervals between the data points. Inother words, the theorem assumes a uniform data signal. Such a theoremis disadvantageous, however, when there is a need to reconstruct anon-uniform data signal. A need exists to overcome this deficiency.

SUMMARY

According to various embodiments, a method for interpolatingnon-uniformly sampled data is provided. The interpolated data can beused to improve the original data. The method can comprise providing anoptical system comprising a signal processor. The method can comprisecollecting data generated by the optical system, wherein the datacomprises non-uniformly sampled data. The method can comprise performingan interpolation operation on the non-uniformly sampled data using thesignal processor, to generate interpolated data. Further, the method cancomprise adjusting the data with the interpolated data using the signalprocessor, to generate improved data. The improved data can be output toa user, for example, by displaying the improved data on a display unit,by printing the improved data, by adjusting an image based on theimproved data, or a combination thereof. According to variousembodiments, the data can comprise any desired data, for example, imagedata. The method can comprise improving the image data. For example, themethod can comprise improving the image resolution, the imagebrightness, the image contrast, and/or the image focus.

According to various embodiments, the interpolation operation cantransform the data from one domain to another. For example, the data canbe transformed from the spatial domain to the frequency domain or fromthe frequency domain to the spatial domain. In some embodiments, theinterpolation operation can comprise transforming the non-uniformlysampled data from the spatial domain to the frequency domain by applyingthe function

${f(x)} = {\frac{2}{Q}{\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}\sin \; {c\left\lbrack {\frac{2}{Q}\left( {n - \frac{x}{\Delta \; x}} \right)} \right\rbrack}}}}$

to the non-uniformly sampled data, wherein ƒ(x) is a sampled periodicfunction, Q is a dimensionless constant for the ratio of a samplingfrequency to a band-limited frequency, n is the sampling interval,

$\sum\limits_{n = {- \infty}}^{+ \infty}$

is the summation from positive infinity to negative infinity, andƒ(x)_(n) is the value of the periodic function at the sampling intervaln.

In some embodiments, the interpolation operation can comprisetransforming the non-uniformly sampled data from the frequency domain tothe spatial domain by applying the function

${F(v)} = {\frac{2}{Q_{v}}{\sum\limits_{n = {- \infty}}^{+ \infty}{{F\left( v_{n} \right)}\sin \; {c\left\lbrack {\frac{2}{Q_{v}}\left( {n - \frac{v}{\Delta \; v}} \right)} \right\rbrack}}}}$

to the non-uniformly sampled data, wherein F(v) is Fourier transform,Q_(v) is a dimensionless constant for the ratio of a sampling intervalto a spatial limit, n is the sampling interval,

$\sum\limits_{n = {- \infty}}^{+ \infty}$

is the summation from positive infinity to negative infinity, andF(v_(n)) is the value of the Fourier transform at the sampling intervaln.

According to various embodiments, the present teachings provide anoptical system that can be configured to perform the method of thepresent teachings. The optical system can comprise a signal processorand a control unit, wherein the control unit is operably linked to thesignal processor. The signal processor can be configured to perform themethod described herein. The signal processor can be configured tocollect image data from the telescope. The image data can comprise atleast one of resolution data, contrast data, brightness data, and focusdata.

BRIEF DESCRIPTION OF THE DRAWINGS

The present teachings will be described with reference to theaccompanying drawings.

FIG. 1 illustrates a Fourier transform of a band-limited sampledfunction and its representation by periodic extension of the samplinginterval, according to various embodiments of the present teachings;

FIG. 2 illustrates the Fourier transform of FIG. 1 with the samplingordinate (−v_(b)) shifted to the origin, according to variousembodiments of the present teachings;

FIG. 3 illustrates neighboring basis functions for Q=2, comprising arapid decay of the side-lobes, according to various embodiments of thepresent teachings;

FIG. 4 illustrates neighboring basis functions for Q=4, comprising arapid decay of the side-lobes, according to various embodiments of thepresent teachings; and

FIG. 5 illustrates the sum of the neighboring basis functions shown inFIG. 4, according to various embodiments of the present teachings.

DETAILED DESCRIPTION OF THE PRESENT INVENTION

According to various embodiments, the Whittaker-Shannon-Kotelnikov (WSK)sampling theorem can be used as a starting point for reconstructing acontinuous band-limited function from a discrete set of sample points.Various references describe the WSK sampling theorem, for example, E. T.Whittaker, “On the Functions which are Represented by the Expansions ofthe Interpolation Theory,” Proc. Royal Soc. Edinburgh, Sec. A, Vol. 35,pp. 181-194 (1915), C. E. Shannon, “Communication in the presence ofnoise,” Proc. Institute of Radio Engineers, vol. 37 (1), pp. 10-21(1949), V. A. Kotelnikov, “On the capacity of the ‘either’ and of cablesin electrical communication.” Procs. Of the 1^(st) All-Union Conferenceon technological reconstruction of the communications sector andlow-current engineering, Moscow (1933), A. J. Jerri, “The ShannonSampling Theorem—Its Various Extensions and Applications: A Tutorialreview,” IEEE Comm., Vol. 65 (11), pp. 1565-1596 (1977), H. D. Luke,“The origins of the sampling theorem,” IEEE comm., Vol. 37 (4), pp.106-108 (1999), and M. Unser, “Sampling—50 Years After Shannon,” IEEEComm., Vol. 99 (4), pp. 569-587, all of which are incorporated herein intheir entireties by reference. The result can be derived using aconvolution theorem in combination with various mathematical computingsoftware, for example, the comb(x) and rect(x) functions of MATLAB®,available from Mathworks Inc., Natick, Mass. The sampling theorem can beused to sample optical fields as described in J. W. Goodman,Introduction to Fourier Optics, 2^(nd) ed. (McGraw Hill, New York, N.Y.,1996), pp. 23-26, and D. A. Aronstein, “Whittacker-Shannon Interpolationof Electric Fields and Point-Spread Functions,” private communication(Apr. 6, 2007), both of which are incorporated herein in theirentireties by reference.

As is known in the art, the WSK sampling theorem assumes a continuousuniformly sampled data signal. Because of this, the WSK sampling theoremis not ideal for use with a non-uniformly sampled data signal. Describedherein is a derivation of the sampling theorem that emphasizes twoassumptions of the theorem explicitly. This derivation achieves apreferred method for interpolating non-uniformly sampled data signals.The theorem can be expressed in terms of two fundamental length scalesthat are derived from these assumptions. The result can be more generalthan what is usually reported and contains the WSK form as a specialcase corresponding to Nyquist sampled data. In some embodiments, thepreferred basis set for interpolation can be found by varying thefrequency component of the basis functions in an optimal way. Thisobservation can give a generalization of the WSK result to situationswhere the data-sampling interval is non-uniform, for example, as isdescribed J. L. Yen, “On nonuniform sampling of band limited signals,”IRE Trans. Circ. Theory CT-3, 251-257 (1956), which is incorporatedherein in its entirety by reference.

According to various embodiments, a method for interpolatingnon-uniformly sampled data is provided. The interpolated data can beused to improve original data. The method can comprise providing anoptical system comprising a signal processor. The method can comprisecollecting data generated by the optical system, wherein the datacomprises non-uniformly sampled data. The method can comprise performingan interpolation operation on the non-uniformly sampled data using thesignal processor, to generate interpolated data. In some embodiments,the method can comprise adjusting the data with the interpolated datausing the signal processor, to generate improved data. The improved datacan be output to a user, for example, by displaying the improved data ona display unit, by printing the improved data, by adjusting an imagebased on the improved data, by a combination thereof, or the like.According to various embodiments, the data can comprise any desireddata, for example, image data. The method can comprise improving theimage data. For example, the method can comprise improving the imageresolution, the image brightness, the image contrast, and/or the imagefocus.

According to various embodiments, the interpolation operation cantransform the data from one domain to another. For example, the data canbe transformed from the spatial domain to the frequency domain or fromthe frequency domain to the spatial domain. The present teachings alsoprovide an optical system that can be configured to perform the methodof the present teachings. The optical system can comprise a signalprocessor and a control unit, where the control unit is operably linkedto the signal processor. The signal processor can be configured toperform the method described herein. In some embodiments, the signalprocessor can be configured to collect image data from a telescope. Theimage data can comprise at least one of resolution data, contrast data,brightness data, and focus data.

Assumptions: Band-Limited Functions and Sampling Interval

According to various embodiments, the Fourier transform can be definedas:

F(v)=ℑ{f(x)}=∫_(−∞) ^(+∞) dxf(x)e ^(−i2πvx)

and its inverse can be represented as:

f(x)=ℑ⁻¹ {F(v)}=∫_(−∞) ^(+∞) dvF(v)e ^(i2πxv)

The sampling theorem can be based on three conditions (A, B, and C)regarding the Fourier transform. The conditions A, B, and C can be:

(A) F(v) can be non-zero over a finite domain, or alternatively stated,its conjugate function ƒ(x) can be band-limited:

F _(b)(v)=ℑ{f(x)}=0 for vε[−v _(b) ,v _(b)]

with |v_(b)|<∞.(B) A periodic function F_(bp)(v) can be constructed from F_(b)(v) byperiodic extension over the data-sampling interval Δx:

F _(bp)(v)=F _(b)(v+nv _(Δ)); for ∀vε° and n=0,1,2, . . .

where v_(Δ)=1/Δx is the data sampling frequency. In some embodiments,condition (A) can imply condition (B) because a periodic function can beconstructed from the F_(b)(v). Therefore, in some embodiments, condition(B) can be implied by (A).(C) Finally, existence of F_(b)(v) can be implied by

∫_(−∞) ^(+∞) dx|f(x)|<∞

or in the discrete case: ƒ_(n)=ƒ(x_(n))=ƒ(nΔx)

According to various embodiments, the band-limited functions shown incondition (A) and condition (B) can imply two fundamental length scalesx_(b)=1/v_(b) and Δx=1/v_(Δ) where x_(n)=nΔx are the data sample points.These length scales, or equivalently their associated frequencyintervals, can be kept independent and play a fundamental role in thederivation of the sampling theorem. According to various embodiments,condition (C) does not hold for periodic functions, for example, sin(x).To overcome this, the generalized Fourier transform for functions of“slow growth” can be used. Discussion on the generalized Fouriertransform can be found in H. P. Hsu, Applied Fourier Analysis, (HarcourtBrace, New York, N.Y., 1984), pp. 98, 104, which is incorporated hereinin its entirety by reference. Parseval's theorem can be used toconstruct the Fourier transform for functions not satisfying condition(C).

According to various embodiments, FIG. 1 illustrates the band-limitedcondition (A) and condition (B) in the frequency domain. In FIG. 1, theFourier transform of a band-limited sampled function is represented byperiodic extension of the sampling interval. Alternatively, the −v_(b)ordinate of FIG. 1 can be shifted to the origin as in FIG. 2. Thisversion of the band-limited sampled function further illustrates theconstruction of condition (B), by periodic extension over the samplingfrequency v_(Δ).

Derivation of the Sampling Theorem

As shown in FIG. 2, condition (B) allows the Fourier transform to beconstructed as a periodic function by periodic extension of the samplingfrequency. Condition (B) does not have to be taken as a fundamentalproperty of the sampled Fourier transform, but can be enforced byconstruction, for example, as:

${F_{bp}(v)} = {{\sum\limits_{n = {- \infty}}^{+ \infty}{c_{n}^{{- {2\pi}}\; {{vn}{({1/v_{\Delta}})}}}}} \equiv {\sum\limits_{n = {- \infty}}^{+ \infty}{c_{n}{^{{2\pi}\; {{nv}{({n\; \Delta \; x})}}}.}}}}$

From condition (A), F_(b)(v)=F_(bp)(v) over ±v_(b), thus ƒ(x) can beexpressed as:

f(x)=∫_(−v) _(b) ^(+v) ^(b) dvF _(b)(v)e ^(i2πvx)

By this interpretation, condition (B) allows for a representation ofƒ(x) as a sampled, but not necessarily periodic, function ƒ(x_(n)). Whenrepresenting F_(bp)(v) in a Fourier sense, as shown in FIG. 2, theNyquist condition can be enforced, for example, v_(Δ)=2v_(b)

Δx=½v_(b). Examples of the Nyquist condition are described in R. W.Hamming, Digital Filters, (3^(rd) Edn, Dover, Mileola, N.Y., 1998). P.174, and H. Nyquist, “Certain topics in telegraph transmission theory,”Trans. AIEE, vol. 47, pp. 617-644 (1928), both of which are incorporatedherein in their entireties by reference. The derivation does not assumea relationship between Δx and v_(b), other than v_(Δ)≧v_(b), thus, Δxand v_(b) can be kept independent. Substituting

${F_{bp}(v)} = {{\sum\limits_{n = {- \infty}}^{+ \infty}{c_{n}^{{- {2\pi}}\; {{vn}{({1/v_{\Delta}})}}}}} \equiv {\sum\limits_{n = {- \infty}}^{+ \infty}{c_{n}^{{2\pi}\; {{nv}{({n\; \Delta \; x})}}}}}}$

into ƒ(x)=∫_(−v) _(b) ^(+v) ^(b) dvF_(b)(v)e^(i2πvx), and theninterchanging the order of summation and integration gives:

${f(x)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{c_{n}{\int_{- V_{b}}^{+ V_{b}}\ {{v}\; {^{{2\pi}\; {v{({x - {n\; \Delta \; x}})}}}.}}}}}$

According to various embodiments, ƒ(x) can be evaluated:

$\begin{matrix}{{f(x)} = {2v_{b}{\sum\limits_{n = {- \infty}}^{+ \infty}{c_{n}\frac{\sin \left\lbrack {2\pi \; {v_{b}\left( {x_{n} - x} \right)}} \right\rbrack}{2\pi \; {v_{b}\left( {x_{n} - x} \right)}}}}}} \\{= {2v_{b}{\sum\limits_{n = {- \infty}}^{+ \infty}{c_{n}\sin \; {c\left\lbrack {2{v_{b}\left( {x_{n} - x} \right)}} \right\rbrack}}}}}\end{matrix}$

where the right hand side can be expressed using the definition of thesin c(x) function:

sin c(x)≡sin(πx)/πx

Solving for the c_(n) in terms of the sampled function values off(x_(n))

c _(n) =f(nΔx)/v _(Δ) =f(x _(n))/v _(Δ)

and then substituting this value for c_(n) into the integral we get thederivation of the sampling theorem in the spatial domain, which can berepresented as:

${f(x)} = {\frac{2}{Q}{\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}\sin \; {{c\left\lbrack {\frac{2}{Q}\left( {n - \frac{x}{\Delta \; x}} \right)} \right\rbrack}.}}}}$

The dimensionless constant Q can be substituted for the ratio of thesampling frequency to the band-limited frequency:

$Q \equiv \frac{v_{\Delta}}{v_{b}}$

Using this substitution, the derivation can be expressed as:

${f(x)} = {2\Delta \; {xv}_{b}{\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}\sin \; {c\left\lbrack {2{v_{b}\left( {x_{n} - x} \right)}} \right\rbrack}}}}$

According to various embodiments, the derivation of the sampling theoremin the spatial domain can provide a method for reconstructing acontinuous band-limited function in terms of the frequency bandwidthv_(b) and the sampling interval Δx. As illustrated in FIGS. 1 and 2 thederivation is valid when:

$\left. {Q \geq 2}\Rightarrow{{\Delta \; x} \leq \frac{1}{2v_{b}}} \right.$

recognizing Q=2 as the Nyquist condition that can be used to avoidaliasing. Substituting the Nyquist condition into the derivation canillustrated that the WSK result can be a special case of the derivationof the sampling theorem:

$\begin{matrix}{{{f(x)}}_{Q\rightarrow 2} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}\sin \; {c\left( {n - {{x/\Delta}\; x}} \right)}}}} \\{= {\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}\sin \; {c\left\lbrack {v_{\Delta}\left( {x_{n} - x} \right)} \right\rbrack}}}}\end{matrix}$

In the Frequency Domain

According to various embodiments, and as described above, the samplingtheorem can be derived in the spatial domain by placing restrictions ona function's Fourier transform and can provide a method forreconstructing a continuous ƒ(x) from its sampled values, ƒ(x_(n)). Insome embodiments, the modulus of the Fourier transform, |F(v)|, can bemeasured rather than the ƒ(x_(n)), so it is of equal interest toreconstruct |F(v)| in the frequency domain, from its sampled valuesusing a formula akin to the derivation of the sampling theorem in thespatial domain. Analogous with condition (A) for the spatial domain,ƒ(x) can be assumed to be spatially limited:

ƒ_(s)(x)=0 for xε[−x _(s) ,x _(s)]

where ƒ_(s)(x) denotes a spatially limited version of ƒ(x). In someembodiments, ƒ(x) is not spatially limited, but ƒ_(s)(x) can beconstructed using an appropriate filtering or windowing operation asdescribed in H. J. Weaver, Applications of Discrete and ContinuousFourier Analysis, (Krieger Publishing Company, Malabar, FI. 1992), pp.134-147, which is incorporated herein in its entirety by reference.

In some embodiments, it is possible to construct a periodic functionƒ_(sp)(x) from ƒ_(s)(x) by periodic extension over the function spatialperiod x_(Δ), for example:

ƒ_(sp)(x)=f(x+nx _(Δ)), for ∀xε° and n=0,1,2, . . .

A spatially limited and periodic representation of f(x) can berepresented by Fourier series as:

${f_{sp}(x)} = {{\sum\limits_{n = {- \infty}}^{+ \infty}{b_{n}^{{2\pi}\; {{xn}{({1/x_{\Delta}})}}}}} \equiv {\sum\limits_{n = {- \infty}}^{+ \infty}{b_{n}^{{2\pi}\; {x{({n\; \Delta \; v})}}}}}}$

where Δv=1/x_(Δ) is a data-sampling interval in the Fourier domain. TheFourier transform of the function can be defined as:

F(v)=∫_(−x) _(S) ^(+x) ^(s) dxƒ _(sp)(x)e ^(−i2πvx)

and the Fourier sense can be substituted into the Fourier transform,while interchanging the order of summation and integration. The resultcan be expressed as:

${F(v)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{b_{n}{\int_{- x_{s}}^{+ x_{s}}\ {{x}\; ^{{2\pi}\; {x{({{n\; \Delta \; v} - v})}}}}}}}$

Evaluating this result can be done as follows:

$\begin{matrix}{{F(v)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{b_{n}\left( {2x_{s}} \right)}\frac{\sin \left\lbrack {2\pi \; {x_{s}\left( {{n\; \Delta \; v} - v} \right)}} \right\rbrack}{2\pi \; {x_{s}\left( {{n\; \Delta \; v} - v} \right)}}}}} \\{= {\sum\limits_{n = {- \infty}}^{+ \infty}{{b_{n}\left( {2x_{s}} \right)}\sin \; {c\left\lbrack {2{L_{s}\left( {{n\; \Delta \; v} - v} \right)}} \right\rbrack}}}}\end{matrix}$

where the right hand side can be expressed using the definition of thesin c(x) function. The variable b_(n) can be expressed in terms of thesampled data values, for example:

b _(n) =F(v _(n))/x _(Δ) =F(nΔv)/x _(Δ).

Substituting b_(n) into F(v), and then rearranging terms, achieves aderivation of the sampling theorem in the frequency domain:

${F(v)} = {\frac{2}{Q_{v}}{\sum\limits_{n = {- \infty}}^{+ \infty}{{F\left( v_{n} \right)}\sin \; {c\left\lbrack {\frac{2}{Q_{v}}\left( {n - \frac{v}{\Delta \; v}} \right)} \right\rbrack}}}}$

where again a dimensionless constant Q_(v) has been substituted for theratio of the sampling interval to the spatial limit (the “v” subscriptdenotes the domain to which it applies):

$Q_{v} \equiv \frac{x_{s}}{x_{\Delta}}$

Alternatively, substituting in the value for Q_(v), the derivation ofthe sampling theorem in the frequency domain can be expressed as:

${F(v)} = {2\Delta \; v\; x_{s}{\sum\limits_{n = {- \infty}}^{+ \infty}{{F\left( v_{n} \right)}{{{sinc}\left\lbrack {2{x_{s}\left( {v_{n} - v} \right)}} \right\rbrack}.}}}}$

The derivations in the frequency domain are analogous to the derivationsin the spatial domain, respectively, for reconstructing the Fouriertransform of a spatially-limited function in terms of the functionspatial limit x_(s) and Fourier sampling interval Δv. Again, thederivation can be valid when:

$\left. {Q_{v} \geq 2}\Rightarrow{{\Delta \; v} \leq \frac{1}{2x_{s}}} \right.$

corresponding to the Nyquist sampling condition.

Arbitrary Domain

Given the definition of the Fourier transfer and its inverse, it ispossible to choose what is labeled as the “spatial (time)” or “Fourier(frequency)” domain. For example, the variables in either domain canarbitrarily be “x and v” or “v and x” as long as the variables areconstant with the given choice and definitions of the Fourier transformand its inverse. Thus, the derivation of the sampling theorem in thefrequency domain can be expected, but its derivation can be instructivein that the spatial limit x_(s) can be seen as the corollary to theband-limit v_(b), and the data sampling intervals, Δx and Δx, can be thecorollaries to their replicated periods, v_(Δ) and X_(Δ), in theconjugate domains, respectively.

Corollary Results

Several corollary results will be discussed in respect to the spatialdomain. According to various embodiments, analogous expressions can bederived in the Fourier domain and in the arbitrary domain using thederivation of the sampling theorem described herein.

Continuous to Discrete, x→x_(m)a

In some embodiments, an alternative statement of the hand-limitedfunction F_(bp)(v)=F_(b)(v+nv_(Δ)) can be examined by considering thelimit of the continuous variable “x” to the discrete case x→x_(m). Inone example, the number of interpolation points can match the number ofdata samples. The sin c term in the sampling theorem derivation can thenbe represented as:

sin c[2v _(b)(x _(n) −x)]|_(x→x) _(m) =sin c[2v _(b)(x _(n) −x_(m))]=sin c[2v _(b) Δx(n−m)]

and using the property of the Dirac delta function that δ(ax)=δ(x)/|a|,the limit can be represented as:

${{{sinc}\left\lbrack {2v_{b}\Delta \; {x\left( {n - m} \right)}} \right\rbrack}_{m->n}} = {\frac{1}{2\; v_{b}\Delta \; x}{{\delta \left( {n - m} \right)}.}}$

Substituting into the band-limited function and allowing the continuousvariable x to become “discretized” to the same number of data samples asthe function, the sin c(x) function interpolator can become the deltafunction:

${f\left( x_{m} \right)} = {\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}{\delta \left( {n - m} \right)}}}$

In some embodiments, the sin c(x) function interpolator can be analternative statement of the main assumption of the band limitedfunction of the periodicity of the Fourier transform. The Fouriertransform of the sin c(x) function interpolator can lead to theband-limited function, which in fact can be a starting assumption whenperforming the derivation of the sampling theorem.

Windowing and the Discrete Form

According to various embodiments, the infinite sum in the samplingtheorem can be substituted with a finite number of data samples N. Thedata can be windowed or truncated which can lead to spectral leakage inthe spatial domain or aliasing in the Fourier domain. Spectral leakageand aliasing of signals can be an undesirable effect, otherwise known asGibbs phenomena, which is described in H. Nyquist, “Certain topics intelegraph transmission theory,” Trans. AIEE, vol. 47, pp. 617-644(1928), which is incorporated herein in its entirety by reference. Themethod of the present teachings can be effective in overcoming orresolving the undesirable effects produced by the Gibbs phenomena.

Digital Representation

According to various embodiments, the numerical examples provided hereincan use a finite representation such that for N data sample points thederivation of the sampling theorem can be represented as:

${f(x)} = {\left( {2/Q} \right){\sum\limits_{n = 0}^{N - 1}{{f\left( x_{n} \right)}{{sinc}\left\lbrack {\left( {2/Q} \right)\left( {{n^{\prime}(n)} - {{x/\Delta}\; x}} \right)} \right\rbrack}}}}$

where it is understood that

${n^{\prime}(n)} = \left\lbrack {{{- \frac{1}{2}}\left( {N - 1} \right)},\ldots \mspace{14mu},{{+ \frac{1}{2}}\left( {N - 1} \right)}} \right\rbrack$

can be the index range over N total points and can include values ofx<0.^(g). In some embodiments, n′ (n) can be of a different index range,for example:

${n^{\prime}(n)} = {\left\lbrack {{{- \frac{1}{2}}N},\ldots \mspace{14mu},{{+ \frac{1}{2}}\left( {N - 1} \right)}} \right\rbrack.}$

Where n′(n)=[−½ (N−1), . . . , +½(N−1)], the index range can compriseone or more of the following qualities:

(a) the range can be symmetric with respect to the interval endpoints,

(b) when N is odd, the range can include the origin n′=0,

(c) when N is odd, the n′ can be integer valued,

(d) when N is odd, there can be an even number of “pixels.”

According to various embodiments, the continuous variable x (theinterpolation points) can be finite, for example, there can be M totalinterpolation points. In some embodiments, M>N, M=N, or M<N. The digitalsampling interval for interpolation, Δx′, can be represented as afraction of the data sampling interval Δx′, for example,

Δx′=x/J,

where J≧1. Therefore, for a given choice off there will be:

M=J(N−1)+1,

interpolation points for x, and thus by analogy with the finiterepresentation, we can specify a free index for the interpolationpoints:

${m \in \left\lbrack {{{- \frac{1}{2}}\left( {M - 1} \right)},\ldots \mspace{14mu},{{+ \frac{1}{2}}\left( {M - 1} \right)}} \right\rbrack} = \left\lbrack {{{- \frac{1}{2}}{J\left( {N - 1} \right)}},\ldots \mspace{14mu},{{+ \frac{1}{2}}{J\left( {N - 1} \right)}}} \right\rbrack$

noting that if N is odd then M will also be odd for integer-valued J.Considering the sin c(x) basis function, we can determine:

${\left( {n^{\prime} - \frac{x}{\Delta \; x}} \right)->\left\lbrack {n^{\prime} - {\left( \frac{m\; \Delta \; x}{J} \right)\frac{1}{\Delta \; x}}} \right\rbrack} = {n^{\prime} - \frac{m}{J}}$

and so the derivation of the sampling theorem in the spatial domain canbe expressed in discrete form as:

${f\left( {x_{m}^{\prime} \equiv {m\; \Delta \; x^{\prime}}} \right)} = {\left( {2/Q} \right){\sum\limits_{n = 0}^{N - 1}{{f\left( x_{n} \right)}{{sinc}\left\lbrack {\left( {2/Q} \right)\left( {n^{\prime} - {m/J}} \right)} \right\rbrack}}}}$

giving a computable realization for the m-th interpolated value of ƒ(x).The discrete form of the WSK result can be represented as:

${{f\left( {x_{m}^{\prime} \equiv {m\; \Delta \; x^{\prime}}} \right)}_{Q->2}} = {\sum\limits_{n = 0}^{N - 1}{{f\left( x_{n} \right)}{{{sinc}\left( {n^{\prime} - {m/J}} \right)}.}}}$

In some embodiments, the number of interpolation points can match thenumber of data points that are sampled. For example, when J=1, thefinite sum can be represented as:

${f\left( x_{m} \right)} = {\sum\limits_{n = 0}^{N - 1}{{f\left( x_{n} \right)}{\delta \left( {n - m} \right)}}}$

Convolution Form

As mentioned above, the sampling theorem in the spatial domain canprovide a method for reconstructing a band-limited function in terms ofthe fundamental length scales: v_(h) and Δx. ƒ(x) can be expressed as aweighted sum over the sin c(x) basis functions. In some embodiments, inthe continuum limit, the band-limited function can become a convolutionintegral:

f(x)=f(x)

sin c(x)=∫_(−∞) ^(+∞) duf(u)sin c(x−u)

The convolution integral can be viewed as an implication of theassumptions in conditions A, B, and C. In continuum form, theimplications of these assumptions can be examined for additional insightinto their analytic structure. The continuum limit of the delta functioncan be represented as:

f(x)=f(x)

δ(x)=∫_(−∞) ^(+∞) duf(u)δ(x−u)

To further emphasize the convolution form, the Fourier transform of theconvolution integral can show that in the frequency domain the integralcan be used to force a finite interval on the Fourier transform of thefunction, and thus a band-limit on ƒ(x):

ℑ{f(x)}=F _(b)(v)=F(v)·rect(v,2v _(b))

According to various embodiments, the convolution integral can be areincarnation of the band-limited assumption and the Fourier transformof the convolution integral can be the Fourier domain representation ofcondition (A). Algebraic examples of the convolution integral can be,for example ƒ(x)=sin c(x), which can be represented as:

${{{sinc}(x)} \otimes {{sinc}(x)}} = {{\int_{- \infty}^{+ \infty}{{{u\left( \frac{\sin \; u}{u} \right)}}\frac{\sin \left\lbrack {\pi \left( {x - u} \right)} \right\rbrack}{\pi \left( {x - u} \right)}}} = {{sinc}(x)}}$

and the result can also be expressed for ƒ(x)=sin c²(x):

${{{sinc}^{2}(x)} \otimes {{sinc}(x)}} = {{\int_{- \infty}^{+ \infty}{{{u\left( \frac{\sin \; u}{u} \right)}^{2}}\frac{\sin \left\lbrack {\pi \left( {x - u} \right)} \right\rbrack}{\pi \left( {x - u} \right)}}} = {{sinc}^{2}(x)}}$

Fourier's Integral Theorem

According to various embodiments, Fourier's Integral Theorem can provideinsight on the convolution form of the convolution integral:

${\int_{0}^{C}{{v}\mspace{11mu} {\cos \left\lbrack {2\pi \; {v\left( {x - u} \right)}} \right\rbrack}}} = {\frac{\sin \left\lbrack {2c\; {\pi \left( {x - u} \right)}} \right\rbrack}{\pi \left( {x - u} \right)} = {c^{\prime}{{sinc}\left\lbrack {c^{\prime}\left( {x - u} \right)} \right\rbrack}}}$

with c′=2c. The result can be:

${\sin \; {c\left\lbrack \left( {x - u} \right) \right\rbrack}} = {\int_{0}^{c^{\prime}}{{v}\; {{\cos \left\lbrack {\frac{1}{c^{\prime}}2\pi \; {v\left( {x - u} \right)}} \right\rbrack}/c^{\prime}}}}$

With a change of integration variable, v/c′→k, and letting k→∞ as v→∞,the result becomes:

sin c[(x−u)]=∫₀ ^(∞) dμ cos [2πμ(x−u)]

Substituting this integral into the convolution integral, andinterchanging the order of integration shows that convolution integralcan be equivalent to:

f(x)=∫_(κ=0) ^(∞)∫_(−∞) ^(+∞) dudkf(u)cos [2πκ(x−u)]

which is Fourier's Integral Theorem. Note that Fourier's IntegralTheorem can also be derived from the definition of the Fourier transformwhen ƒ(x) is real. The forward and inverse Fourier transforms arerepresented as:

F(v)=∫_(−∞) ^(+∞) duf(x)e ^(−2πvu) and f(x)=∫_(−v) _(b) ^(+v) ^(b) dvF_(b)(v)e ^(i2πvx)

Substituting the value of ƒ(v) into ƒ(x) gives:

$\begin{matrix}{{f(x)} = {\int_{- \infty}^{+ \infty}{{{v\left\lbrack {\int_{- \infty}^{+ \infty}{{{{uf}(u)}}^{{- {2\pi}}\; {vu}}}} \right\rbrack}}^{{2\pi}\; {vx}}}}} \\{= {\int_{- \infty}^{+ \infty}{\int_{- \infty}^{+ \infty}{{v}{{{uf}(u)}}{^{{2\pi}\; {v{({x - u})}}}.}}}}}\end{matrix}$

Assuming ƒ(x) is real:

ƒ_(R)(x)=f(x)=∫_(−∞) ^(+∞)∫_(−∞) ^(+∞) dvduƒ(u)cos [2πv(x−u)]

but because cos [2πv(x−u)] is even with respect to v, the equation canbe written as

ƒ_(R)(x)=f(x)=∫₀ ^(+∞)∫_(−∞) ^(+∞) dvduƒ(u)cos [2πv(x−u)]

which is the Fourier Integral Theorem.

Optimal Basis Function Frequency

According to various embodiments, the sampling theorem in abbreviatedform can be represented as:

${{f(x)} \propto {\sum\limits_{n}{{f\left( x_{n} \right)}\sin \; {c\left\lbrack {\alpha \; {u(n)}} \right\rbrack}}}};\mspace{14mu} {u \equiv \left( {n - {{x/\Delta}\; x}} \right)}$

which can help to emphasize another way of looking at the problem,noting that the coefficient α plays the role of a modulating frequency.The coefficient α can determine the spacing for the subsidiaryside-lobes of sin c(αu), for example, the various zeros, maxima, andminima of the sin c(αu) basic functions. The zeros can be given by

παu(n)=πk

u(n)=k/α; for k=1,2,3, . . .

As discussed below, if α≠1, the zeros of sin c(αu) do not coincide withthe data sampling interval Δx.

In some embodiments, ƒ(x) can be a superposition of displaced andweighted sin c(αu) functions, therefore, cancellation of the subsidiaryside-lobes at the base of the sin c(αu), can improve interpolationresults. To further elaborate, consider an example where Q=4, and forconvenience, what is produced is:

Q=4;v _(b)=2

x _(b)=1;Δx=⅛

v _(Δ)=8

The WSK result corresponds to α=1 and the two terms of this example(terms n=20, n+1=21, and N=41) are plotted on a graph and shown in FIG.3, where it seen that the zeros of the sin c(αu) basis functionscorrespond to the data sampling interval, Δx. This means that whenshifting the neighboring basis functions over by one data unit to thenext, the side-lobe maxima of the n-th basis function correspondsapproximately to the side-lobe minima of the (n+1)-th basis function.Therefore, aside from the interpolated value of ƒ(x) at about (n+½)Δx,the remaining contribution to the sum of the two basis functionsapproximately cancels because the side-lobes are out of phase. The sumof the two basis functions can be, for example, as shown in FIG. 3 wherethe rapid decay of the side-lobes is apparent.

According to various embodiments, by contrast to the WSK result withα=1, the WSK(Q) result has α=½ when Q=4. In this case, the zeros of thesin c(αu) functions are located 2k apart for k given when

παu(n)=πk

u(n)=k/α; for k=1, 2, 3, . . . therefore, the sin c(αu) zeros go throughevery other data point. As a result, when shifting the basis functionover by one data unit in the interpolation sum, the basis functionside-lobes can be approximately “in phase” and little cancellation canoccur in the sum. The situation is illustrated in FIG. 4 along with thesum of the two basic functions. As a result, the WSK(Q) result canprovide a more efficient method of interpolating the non-uniformlysampled data. The WSK(Q) result can be optimal in cases where the datasampling interval is non-constant, and an optimal superposition of thevarious neighboring basis function can be achieved by varying the αcoefficient in a way that minimizes the subsidiary side-lobes.

According to various embodiments, optimally out of phase describeschoices of a that minimize ringing in the subsidiary side-lobes becausein such cases a better overall interpolation result is obtained. Thesummation shown in FIG. 5 is of interest from another standpoint: itappears similar in form to the two-lobed Lanczos-windowed sin cfunction:

${{Lanczos}\; 2\left( {\alpha \; u} \right)} = \left\{ \begin{matrix}{{\sin \; {c\left( {\alpha \; u} \right)}\sin \; {c\left( {\frac{1}{2}\alpha \; u} \right)}},} & {{x} < 2} \\{0,} & {{x} \geq 2.}\end{matrix} \right.$

It is clear that a parallel with the windowing analysis is more thanjust coincidence, and in fact provides additional insight into the WSKinterpolation results.

In some embodiments, shifting the sin c(αu) basis functions so thattheir zeros coincide, can achieve optimal shifting of the sin c(αu)basis functions for u with α=1. For some values of x, this trade is giveand take because some ringing in the sum is unavoidable because thelocations of the maxima of sin c(αu) vary non-linearly with x. This isbecause the maxima are solutions to the transcendental equation:

tan(αu)−αu=0.

For

${{\alpha \; u}\frac{\pi}{2}},$

the solutions to the above equation can be represented as (which can beverified by plotting tan(αu) and αu):

${{{\tan \left( {\alpha \; u} \right)} - {\alpha \; u{\quad }_{{\alpha \; u}\frac{\pi}{2}}}} = {\left. 0\Rightarrow{\alpha \; u} \right. = {\left( {{2p} + 1} \right)\frac{\pi}{2}}}},\mspace{14mu} {p = 1},2,3,\ldots$

For smaller values of α u the maxima of sin c(αu) are:

${{{\tan \left( {\alpha \; u} \right)} - {\alpha \; u}} = {\left. 0\Rightarrow{\alpha \; u} \right. = {{\left( {{2p} + 1} \right)\frac{\pi}{2}} - {ɛ(u)}}}},\mspace{14mu} {p = 1},2,3,\ldots$

where ε(u) is a non-linear function that goes to zero for large u.

Other embodiments will be apparent to those skilled in the art fromconsideration of the present specification and practice of variousembodiments disclosed herein. It is intended that the presentspecification and examples be considered as exemplary only.

What is claimed is:
 1. A method for improving image data, the methodcomprising: providing an optical system comprising a signal processor;collecting image data generated by the optical system, wherein the imagedata comprises non-uniformly sampled data; performing an interpolationoperation on the non-uniformly sampled data using the signal processor,to generate interpolated data; adjusting the image data with theinterpolated data using the signal processor, to generate improved imagedata; and outputting the improved image data to a user.
 2. The method ofclaim 1, wherein the interpolation operation comprises transforming thenon-uniformly sampled data from the spatial domain to the frequencydomain by applying the function${{f(x)} = {\frac{2}{Q}{\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}\sin \; {c\left\lbrack {\frac{2}{Q}\left( {n - \frac{x}{\Delta \; x}} \right)} \right\rbrack}}}}},$to the non-uniformly sampled data, wherein ƒ(x) is a sampled periodicfunction, Q is a dimensionless constant for the ratio of a samplingfrequency to a band-limited frequency, n is the sampling interval,$\sum\limits_{n = {- \infty}}^{+ \infty}$ is the summation from positiveinfinity to negative infinity, and ƒ(x)_(n) is the value of the periodicfunction at the sampling interval n.
 3. The method of claim 1, whereinthe interpolation operation comprises transforming the non-uniformlysampled data from the frequency domain to the spatial domain by applyingthe function${F(v)} = {\frac{2}{Q_{v}}{\sum\limits_{n = {- \infty}}^{+ \infty}{{F\left( v_{n} \right)}\sin \; {c\left\lbrack {\frac{2}{Q_{v}}\left( {n - \frac{v}{\Delta \; v}} \right)} \right\rbrack}}}}$to the non-uniformly sampled data, wherein F(v) is Fourier transform,Q_(v) is dimensionless constant for the ratio of a sampling interval toa spatial limit, n is the sampling interval,$\sum\limits_{n = {- \infty}}^{+ \infty}$ is the summation from positiveinfinity to negative infinity, and F(v_(n)) is the value of the Fouriertransform at the sampling interval n.
 4. The method of claim 1, whereinthe outputting the improved image data to a user comprises at least oneof displaying the improved image data on a display unit and printing outthe improved image data.
 5. The method of claim 1, wherein the improvedimage data comprises improved image resolution data.
 6. The method ofclaim 1, wherein the improved image data comprises at least one ofimproved image contrast data, improved image brightness data, andimproved image focus data.
 7. The method of claim 1, wherein the opticalsystem comprises a telescope and the image data is collected from thetelescope.
 8. An optical system comprising a signal processor and acontrol unit, the control unit being operably linked to the signalprocessor, wherein the signal processor is configured to perform amethod comprising: collecting image data generated by the opticalsystem, wherein the image data comprises non-uniformly sampled data;performing an interpolation operation on the non-uniformly sampled datausing the signal processor, to generate interpolated data; adjusting theimage data with the interpolated data to generate improved image data;and outputting the improved image data to a user.
 9. The optical systemof claim 8, wherein the interpolation operation comprises transformingthe non-uniformly sampled data from the spatial domain to the frequencydomain by applying the function${f(x)} = {\frac{2}{Q}{\sum\limits_{n = {- \infty}}^{+ \infty}{{f\left( x_{n} \right)}\sin \; {c\left\lbrack {\frac{2}{Q}\left( {n - \frac{x}{\Delta \; x}} \right)} \right\rbrack}}}}$to the non-uniformly sampled data, wherein ƒ(x) is a sampled periodicfunction, Q is a dimensionless constant for the ratio of a samplingfrequency to a band-limited frequency, n is the sampling interval,$\sum\limits_{n = {- \infty}}^{+ \infty}$ is the summation from positiveinfinity to negative infinity, and ƒ(x)_(n) is the value of the periodicfunction at the sampling interval n.
 10. The optical system of claim 8,wherein the interpolation operation comprises transforming thenon-uniformly sampled data from the frequency domain to the spatialdomain by applying the function${F(v)} = {\frac{2}{Q_{v}}{\sum\limits_{n = {- \infty}}^{+ \infty}{{F\left( v_{n} \right)}\sin \; {c\left\lbrack {\frac{2}{Q_{v}}\left( {n - \frac{v}{\Delta \; v}} \right)} \right\rbrack}}}}$to the non-uniformly sampled data, wherein F(v) is Fourier transform,Q_(v) is dimensionless constant for the ratio of a sampling interval toa spatial limit, n is the sampling interval,$\sum\limits_{n = {- \infty}}^{+ \infty}$ is the summation from positiveinfinity to negative infinity, and F(v_(n)) is the value of the Fouriertransform at the sampling interval n.
 11. The optical system of claim 8,further comprising a telescope in communication with the control unit,wherein the image data that the signal processor is configured tocollect comprises telescope image resolution data.
 12. The opticalsystem of claim 8, wherein the image data that the signal processor isconfigured to collect comprises at least one of image contrast data,image brightness data, and image focus data.
 13. The optical system ofclaim 8, wherein the image data that the signal processor is configuredto collect comprises image resolution data.
 14. The optical system ofclaim 8, further comprising a telescope in data-transfer communicationwith the control unit, wherein the signal processor is configured tocollect telescope image data.